curve
Summary.
The term curve is associated with two closely related notions. The first notion is kinematic: a parameterized curve is a function of one real variable taking values in some ambient geometric setting. This variable can be interpreted as time, in which case the function describes the evolution of a moving particle. The second notion is geometric; in this sense a curve is an arc, a 1-dimensional subset of an ambient space. The two notions are related: the image of a parameterized curve describes the trajectory of a moving particle. Conversely, a given arc admits multiple parameterizations. A trajectory can be traversed by moving particles at different speeds.
In algebraic geometry, the term curve is used to describe a
1-dimensional variety
relative to the complex numbers or some other
ground field. This can be potentially confusing, because a curve over
the complex numbers refers to an object which, in conventional
geometry
, one would refer to as a surface
. In particular, a Riemann
surface can be regarded as as complex curve.
Kinematic definition
Let be an interval (http://planetmath.org/Interval) of the real line. A parameterized
curve is a continuous mapping taking values in a
topological space
. We say that is a simple curve
if it has no self-intersections, that is if the mapping is
injective
.
We say that is a closed curve, or a loop (http://planetmath.org/loop) whenever is a closed interval, and the endpoints are mapped to the same value; Equivalently, a loop may be defined to be a continuous mapping whose domain is the unit circle. A simple closed curve is often called a Jordan curve.
If then is called a plane curve or planar curve.
A smooth closed curve in is locally
if the local multiplicity of intersection
of with each hyperplane
at of each of the intersection points does not
exceed . The global multiplicity is the sum of the local
multiplicities.
A simple smooth curve in is called (or
globally ) if the global multiplicity
of its intersection with any affine hyperplane is less than or equal to .
An example of a closed convex curve in is the normalized
generalized ellipse:
In odd dimension there are no closed convex curves.
In many instances the ambient space is a differential manifold, in
which case we can speak of differentiable curves. Let be an open
interval, and let be a differentiable curve. For
every can regard the derivative
(http://planetmath.org/RelatedRates),
, as the velocity (http://planetmath.org/RelatedRates) of a
moving particle, at time . The velocity is a
tangent vector
(http://planetmath.org/TangentSpace), which belongs to
, the tangent space
of the manifold at the point
. We say that a differentiable curve is
regular
, if its velocity, , is non-vanishing
for all .
It is also quite common to consider curves that take values in
. In this case, a parameterized curve can be regarded as a
vector-valued function , that is an
-tuple of functions
where , are scalar-valued functions.
Geometric definition.
A (non-singular) curve , equivalently, an arc, is a connected,
1-dimensional submanifold
of a differential manifold . This means
that for every point there exists an open neighbourhood
of and a chart such that
for some real .
An alternative, but equivalent definition, describes an arc as the
image of a regular parameterized curve. To accomplish this, we need
to define the notion of reparameterization. Let be intervals. A reparameterization is a continuously
differentiable function
whose derivative is never vanishing. Thus, is either monotone increasing, or monotone decreasing. Two regular, parameterized curves
are said to be related by a reparameterization if there exists a reparameterization such that
The inverse of a reparameterization
function is also a reparameterization. Likewise, the composition of
two parameterizations is again a reparameterization. Thus the
reparameterization relation
between curves, is in fact an
equivalence relation. An arc can now be defined as an equivalence
class
of regular, simple curves related by reparameterizations. In
order to exclude pathological embeddings
with wild endpoints we also
impose the condition that the arc, as a subset of , be
homeomorphic to an open interval.
Title | curve |
Canonical name | Curve |
Date of creation | 2013-03-22 12:54:17 |
Last modified on | 2013-03-22 12:54:17 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 28 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 53B25 |
Classification | msc 14H50 |
Classification | msc 14F35 |
Classification | msc 51N05 |
Synonym | parametrized curve |
Synonym | parameterized curve |
Synonym | path |
Synonym | trajectory |
Related topic | FundamentalGroup |
Related topic | TangentSpace |
Related topic | RealTree |
Defines | closed curve |
Defines | Jordan curve |
Defines | regular curve |
Defines | simple closed curve |
Defines | simple curve |
Defines | plane curve |
Defines | planar curve |
Defines | convex curve |
Defines | locally convex curve |
Defines | local multiplicity |
Defines | globally convex |
Defines | global multiplicity |